3.1. Between the completion of his 1814 memoir and his next major contribution to complex function theory, the famous memoir [1825b] on definite integrals between imaginary limits, Cauchy made a number of minor improvements to the theory; in addition, because the publication of the 1814 memoir was held up until 1827, he restated his main results several times, in particular in the papers [1822b] and [1823a] and in his book [1823b] on the infinitesimal calculus.

To get the chronology right, it has to be noted that [1823a] falls into two parts, arising, respectively, from memoirs presented to the Academy of Sciences on 16 September 1822 and 26 May 1823; although the memoir was published in July 1823 and the Calcul infinitésimal [1823b] in August, the second part of [1823a] contains references to [1823b]. It is this second part with which we shall mainly be concerned.

We shall also have to take into account some of the supplementary notes to his prize memoir [1815] on water waves, which appear to date from about 1824, and the footnotes that he prepared about 1825 for insertion in the published version of the 1814 memoir. We shall present some evidence about the contents of his 1817 lectures at the Collège de France and an unpublished memoir summarised in [1819]; some relevant material in other papers of this period and in the Analyse algébrique will also be mentioned.